JournalscmhVol. 74 , No. 2DOI 10.1007/s000140050088

The singly periodic genus-one helicoid

  • D. Hoffman

    Berkeley, USA
  • H. Karcher

    Universität Bonn, Germany
  • F. Wei

    Virginia Tech, Blacksburg, USA
The singly periodic genus-one helicoid cover

Abstract

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: 180180^\circ -rotation about the vertical line; 180180^\circ rotation about the horizontal lines (the same symmetry); and their composition.